Multipartite Ramsey numbers

For a graph G, a partiteness k ≥ 2 and a number of colours c, we define the multipartite Ramsey number r^c_k(G) as the minimum value m such that, given any colouring using c colours of the edges of the complete balanced k-partite graph with m vertices in each partite set, there must exist a monochromatic copy of G. We show that the question of the existence of r^c_k(G) is tied up with what monochromatic subgraphs are forced in a c-colouring of the complete graph K_k. We then calculate the values for some small G including r²₃ (C₄) = 3, r²₄(C₄) = 2, r³₃(C₄) = 7 and r²₃(C₆) = 3.